# American Institute of Mathematical Sciences

October  2019, 12(6): 1635-1668. doi: 10.3934/dcdss.2019111

## On the controllability of racing sailing boats with foils

 1 Département d'ingénierie mathématique, laboratoire M2N, 292, rue saint Martin, 75003 Paris, France 2 Laboratoire de mathématiques d'Orsay, UMR 8628, Univ Paris-Sud, CNRS, Université Paris-Saclay, Orsay 91405, France

Received  November 2017 Revised  April 2018 Published  November 2018

The development of foils for racing boats has changed the strategy of sailing. Recently, the America's cup held in San Francisco, has been the theatre of a tragicomic history due to the foils. During the last round, the New-Zealand boat was winning by 8 to 1 against the defender USA. The winner is the first with 9 victories. USA team understood suddenly (may be) how to use the control of the pitching of the main foils by adjusting the rake in order to stabilize the ship. And USA won by 9 victories against 8 to the challenger NZ. Our goal in this paper is to point out few aspects which could be taken into account in order to improve this mysterious control law which is known as the key of the victory of the USA team. There are certainly many reasons and in particular the cleverness of the sailors and of all the engineering team behind this project. But it appears interesting to have a mathematical discussion, even if it is a partial one, on the mechanical behaviour of these extraordinary sailing boats. The numerical examples given here are not the true ones. They have just been invented in order to explain the theoretical developments concerning three points: the possibility of tacking on the foils for sailing upwind, the nature of foiling instabilities, if there are, when the boat is flying and the control laws.

Citation: Philippe Destuynder, Caroline Fabre. On the controllability of racing sailing boats with foils. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1635-1668. doi: 10.3934/dcdss.2019111
##### References:
 [1] R. Bellman, Dynamic Programming, Dover Publications, Inc., Mineola, NY, 2003. [2] H. Brezis, Analyse Fonctionnelle, edition Masson, Paris, 1983. [3] J. Cea, Optimisation, Théorie et Algorithmes, Dunod, Paris, 1971. [4] P.-G. Ciarlet, Introduction à L'analyse Numérique et à L'optimisation, Masson, Paris, 1982. [5] Ph. Destuynder, Introduction à L'aéroélasticité et à L'aéroacoustique, Hermès-Lavoisier, Paris-Londres, 2008. [6] Ph. Destuynder, Analyse et Contrôle des Équations Différentielles, Hermès-Lavoisier, Paris-Londres, 2010. [7] Ph. Destuynder and C. Fabre, Sailing boats with foils, To appear, 2017. [8] Ph. Destuynder and M. T. Ribereau, Non linear dynamics of test models in wind tunnels, Eur. J. Mech. A/Solids, 15 (1996), 91-136. [9] E. H. Dowell, H. C. Curtiss Jr., R. H. Scanlan and F. Sisto, A Modern Course in Aeroelasticity, Monographs and textbooks of solids and fluids. Alphen aan den Rijn, Sijthoff and Noordhoff International Publishers, 1978. [10] A. Ducoin and Y.-L. Young, Hydroelastic response and stability of a hydrofoil in viscous flow, in Journ. of Fluids and Structures, 38 (2013), 40–57. [11] Y. C. Fung, An Introduction to the Theory of Aeroelasticity, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1955. [12] A. J. Hermans, G. C. Hsiao and R. Timman, Water Waves and Ship Hydrodynamics, Delft University Press, The Netherlands, 1985. doi: 10.1007/978-94-017-3657-2. [13] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Masson, Paris, 1988. [14] J. L. Lions, Perturbations Singulères dans les Problèmes Aux Limites et en Contrôle Optimal, Springer-Verlag, Berlin-New York, 1973.

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##### References:
 [1] R. Bellman, Dynamic Programming, Dover Publications, Inc., Mineola, NY, 2003. [2] H. Brezis, Analyse Fonctionnelle, edition Masson, Paris, 1983. [3] J. Cea, Optimisation, Théorie et Algorithmes, Dunod, Paris, 1971. [4] P.-G. Ciarlet, Introduction à L'analyse Numérique et à L'optimisation, Masson, Paris, 1982. [5] Ph. Destuynder, Introduction à L'aéroélasticité et à L'aéroacoustique, Hermès-Lavoisier, Paris-Londres, 2008. [6] Ph. Destuynder, Analyse et Contrôle des Équations Différentielles, Hermès-Lavoisier, Paris-Londres, 2010. [7] Ph. Destuynder and C. Fabre, Sailing boats with foils, To appear, 2017. [8] Ph. Destuynder and M. T. Ribereau, Non linear dynamics of test models in wind tunnels, Eur. J. Mech. A/Solids, 15 (1996), 91-136. [9] E. H. Dowell, H. C. Curtiss Jr., R. H. Scanlan and F. Sisto, A Modern Course in Aeroelasticity, Monographs and textbooks of solids and fluids. Alphen aan den Rijn, Sijthoff and Noordhoff International Publishers, 1978. [10] A. Ducoin and Y.-L. Young, Hydroelastic response and stability of a hydrofoil in viscous flow, in Journ. of Fluids and Structures, 38 (2013), 40–57. [11] Y. C. Fung, An Introduction to the Theory of Aeroelasticity, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1955. [12] A. J. Hermans, G. C. Hsiao and R. Timman, Water Waves and Ship Hydrodynamics, Delft University Press, The Netherlands, 1985. doi: 10.1007/978-94-017-3657-2. [13] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Masson, Paris, 1988. [14] J. L. Lions, Perturbations Singulères dans les Problèmes Aux Limites et en Contrôle Optimal, Springer-Verlag, Berlin-New York, 1973.
Principle of the flying boat with foils
Two situations where the controlled foiling could avoid to tack and to loose the race
The boat with the foils
Heaving exact control with null initial pitching and $u = 10m/s$
Heaving exact control with null initial pitching and $u = 15m/s$
Heaving exact control with nul initial pitching and $u = 20m/s$
Pitching exact control with nul initial heaving and $u = 10m/s$. One can observe that the control is very close to bang-bang control (first negative and then positive. It is almost zero after $t = 3$ sec
Pitching exact control with nul initial heaving and $u = 15m/s$
Pitching exact control with nul initial heaving and $u = 20m/s$
Control with contraints and an initial heaving with null initial pitching and velocity $u = 10m/s$. One can notice that at the beginning the control is bang-bang and finally the adjustement is interior to the upper bounds of the control. Even if the computations are performed with $T = 10$ sec.; one can see that the exact controllability is quite performed at $t = 5$ sec. with the values chosen which are realistic
Control with contraints and an initial heaving with null initial pitching and velocity $u = 15m/s$
Control with contraints and an initial heaving with null initial pitching and velocity $u = 20m/s$
Control with contraints and an initial pitching with null initial heaving and velocity $u = 10m/s$
Control with contraints with an initial pitching with null initial heaving and velocity $u = 15m/s$
Control with contraints with initial pitching with null initial heaving and velocity $u = 20m/s$
The notations used in this section
Linear damping of the model (44) (minus the imaginary part of the eigenvalues $\lambda$) on the top Figure and the frequencies (real part of $\lambda$ divided by $2\pi$) on the bottom one, versus the velocity of the ship ($m./sec.)$
Several trajectories for different values of $\alpha_{max}$ starting from the same initial condition. One can observe that $\alpha_{max}$ should be adjusted in order to obain the right control. This is why the helmsman has a control box which enables him to adjust $\alpha_{max}$ by steps of $\pm.5^0$
Several trajectories for different values of $\alpha_{max}$ starting from the same initial condition. One can observe that this control is unuseful. This is why the rocker switcher has to detect the sign of the velocity of the rake and not the one of the angle (the pitching angle)
OTUSA control of the heaving with velocity 10 m/s
OTUSA control of the heaving with velocity 15 m/s
OTUSA control of the heaving with velocity 20 m/s
OTUSA control of the pitching with velocity 10 m/s
OTUSA control of the pitching with velocity 15 m/s
OTUSA control of the pitching with velocity 20 m/s
The Oracle USA-Team used this mechanical control to win the last eight races in a row. In particular, but not only, it enables them to navigate in the wake of the challenger mainly at the tacking
Principle of the device used published on internet by ORACLE
Expressions of the coefficients $C_{ij}$ near $\gamma = 0$ versus $\alpha$ and $\beta$
 Coefficient Expression of the coefficients of the matrix ${\mathcal C}$ near $\gamma=0$ $C_{11}$ $- \frac{\rho _e}{2} u \bigg (S_s R_{zs}+S_fR_{zf}\bigg )$ $C_{12}$ $\rho _e u \bigg (S_s d_sc_{zs}^0+S_fd_f\cos(\alpha^0)c_{zf}+\frac{S_sh}{2}R_{zs}-\frac{S_fd_f}{2}\sin(\alpha^0)R_{zf}\bigg )$ $C_{21}$ $\frac{\rho _e u}{2} \bigg (-S_s LR_{ms}-LS_fR_{mf}-S_fd_f\sin(\alpha^0)R_{zf} +S_shR_{zs}\bigg )$ $C_{22}$ $\begin{array}{l} -\frac{\varrho_eS_shu}{2}[2d_s c_{zs}^0+hR_{zs}]+\frac{\varrho_euS_fd_f^2}{2}[\sin(2\alpha^0)c_{zf}^0-\sin(\alpha^0)^2 R_{zf}]\\+\frac{\varrho_e u LS_s}{2}\big[2d_sc_{ms}^0+hR_{ms}\big]+\frac{\varrho_e u L}{2}S_fd_f\big[-\sin(\alpha^0)R_{mf}+2\cos(\alpha^0)c_{mf}^0\big]\end{array}$
 Coefficient Expression of the coefficients of the matrix ${\mathcal C}$ near $\gamma=0$ $C_{11}$ $- \frac{\rho _e}{2} u \bigg (S_s R_{zs}+S_fR_{zf}\bigg )$ $C_{12}$ $\rho _e u \bigg (S_s d_sc_{zs}^0+S_fd_f\cos(\alpha^0)c_{zf}+\frac{S_sh}{2}R_{zs}-\frac{S_fd_f}{2}\sin(\alpha^0)R_{zf}\bigg )$ $C_{21}$ $\frac{\rho _e u}{2} \bigg (-S_s LR_{ms}-LS_fR_{mf}-S_fd_f\sin(\alpha^0)R_{zf} +S_shR_{zs}\bigg )$ $C_{22}$ $\begin{array}{l} -\frac{\varrho_eS_shu}{2}[2d_s c_{zs}^0+hR_{zs}]+\frac{\varrho_euS_fd_f^2}{2}[\sin(2\alpha^0)c_{zf}^0-\sin(\alpha^0)^2 R_{zf}]\\+\frac{\varrho_e u LS_s}{2}\big[2d_sc_{ms}^0+hR_{ms}\big]+\frac{\varrho_e u L}{2}S_fd_f\big[-\sin(\alpha^0)R_{mf}+2\cos(\alpha^0)c_{mf}^0\big]\end{array}$
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